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A Universal Identity for Powers in Quadratic Algebras and a Matrix Derivation of a Fibonacci Identity

Marco Mantovanelli

Abstract

We prove a universal identity for powers of elements in quadratic algebras, expressing x^m in terms of x and the identity. As a consequence, we obtain a general formula for powers of 2x2 matrices depending only on trace and determinant. Applying this to the Fibonacci matrix yields a binomial expansion formula for F_{nm}, recovering a recent identity of Vorobtsov. This shows that such identities arise from general algebraic principles rather than specific properties of Fibonacci numbers.

A Universal Identity for Powers in Quadratic Algebras and a Matrix Derivation of a Fibonacci Identity

Abstract

We prove a universal identity for powers of elements in quadratic algebras, expressing x^m in terms of x and the identity. As a consequence, we obtain a general formula for powers of 2x2 matrices depending only on trace and determinant. Applying this to the Fibonacci matrix yields a binomial expansion formula for F_{nm}, recovering a recent identity of Vorobtsov. This shows that such identities arise from general algebraic principles rather than specific properties of Fibonacci numbers.
Paper Structure (6 sections, 4 theorems, 21 equations)

This paper contains 6 sections, 4 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Universal quadratic identity
  3. Matrix formulation
  4. Remark on Chebyshev polynomials
  5. Application to Fibonacci numbers
  6. Conclusion

Key Result

Theorem 1

Let $R$ be a commutative ring with identity, and let $x$ be an element of an $R$-algebra satisfying for some $t,d\in R$. Define polynomials $P_m(t,d)$ by and Then for every integer $m\ge1$, Moreover,

Theorems & Definitions (6)

  • Theorem 1: Universal quadratic reduction
  • proof
  • Proposition 1
  • Corollary 1
  • proof
  • Corollary 2